The Exactly Solvable Simplest Model for Queue Dynamics

We present an exactly solvable model for queue dynamics. Our model is very simple but provides the essential property for such dynamics. As an example, the model has the traveling cluster solution as well as the homogeneous flow solution. The model is the limiting case of Optimal Velocity (OV) model, which is proposed for the car following model to induce traffic jam spontaneously. The concept of queue dynamics offers simple 1-dimensional models for socioeconomic and complex multi-body physical systems, such as traffic and granular transport problems. Recently, several approaches have been developed on this field [1, 2, 3, 5, 7]. We propose an exactly solvable simplest model of this kind. Our model describes the general aspects of queue dynamics, and it can be widely applicable to such systems. But the following discussion is presented with the terminology of traffic problems. The model is ẍn = a {V (∆xn)− ẋn} , (1) where ∆xn = xn−1 − xn, (2) for each car number n (n = 1, 2, · · · ). xn is the position of the n-th car, ∆xn is the headway of that car. Dot denotes the time derivative. a is a sensitivity constant, which we set the same value for all drivers. The function V (∆xn), which is called “OV-function”, is V (∆xn) = vmax · θ(∆xn − d), (3) where θ is a Heaviside function. It decides the optimal velocity (the safety velocity) according to the headway: (a) if the headway is less than d, a car e-mail address: [email protected] e-mail address: [email protected]